Acoustic streaming around a spherical microparticle/cell under ultrasonic wave excitation
|
|
- Agnes Parsons
- 5 years ago
- Views:
Transcription
1 Acoustic streaming around a spherical microparticle/cell under ultrasonic wave excitation Zhongheng Liu a) Yong-Joe Kim b) Acoustics and Signal Processing Laboratory, Department of Mechanical Engineering, Texas A&M University College Station, Texas, TX, , USA A microparticle/cell under an ultrasonic wave excitation generates acoustic streaming due to nonlinear interaction between the particle surface vibration and the wave excitation. For a compact spherical particle whose sie is much smaller than the excitation wavelength, an analytical approach has been developed for the calculation of the streaming. However, for a particle whose sie is comparable to or larger than the wavelength, existing methods require large computational resources and their results are inaccurate in the farfield of the particle. Here, a novel method is proposed to accurately and computationally efficiently calculate the streaming around particles of any sies. In this approach, the acoustic streaming is decomposed into the compressional and shear components based on the viscous boundary layer around the particle. Accordingly, the governing equations of the acoustic streaming are then divided into homogeneous and particular ones depending on the compressional and shear streaming components, respectively. Then, each set of the equations is solved and the total solution is obtained by combining the compressional and shear acoustic streaming solutions. This method is validated by comparing its results with those obtained from the existing analytical method for compact-sie particles. INTRODUCTION The acoustic streaming generated around a microparticle/cell is a nonlinear phenomenon induced by the interaction of the microparticle/cell with an ultrasonic wave excitation in a fluid medium. For the calculation of the acoustic streaming, the interaction between the microparticle/cell and its surrounding fluid medium is decomposed into both linear and nonlinear components. Then, the linear oscillation of the fluid medium is analyed first. Based on the linear oscillation results, the time-averaged nonlinear oscillation of the fluid medium, i.e., the acoustic streaming is then calculated. However, the computation of the acoustic streaming has been a challenging problem and investigated for more than a half century. a) liu008@tamu.edu b) joekim@tamu.edu
2 Raney et al. and Holtsmark et al. 2 studied the acoustic streaming generated around an infinite-length cylinder under an acoustic wave excitation. Later, Lane 3 modified the latter approach to be applied for the calculation of the acoustic streaming generated around a sphere. In this work, a vorticity equation was solved in an incompressible viscous fluid medium with the solution of a stream function by using a perturbation method and the spherical axisymmetric condition. Nyborg 4 developed a general method to calculate acoustic streaming near an arbitrarily-shaped boundary with the assumption of the thin acoustic boundary layer. Wang 5 analyed the acoustic streaming after decomposing it into inner and outer ones of an acoustic boundary layer generated on a spherical boundary. Riley 6 also calculated the inner and outer streaming around a sphere from the vorticity equation. Lee and Wang applied the Nyborg s method for the calculation of the acoustic streaming pattern, near a small sphere, induced by two orthogonal standing waves at a single frequency. 7 They also applied the same method to calculate the outer streaming around a sphere placed between the pressure node and antinode of a single standing wave. 8 Later, Sadhal et al. 9,0 studied the acoustic streaming around a droplet, gas bubble, and solid elastic particle. All of the aforementioned approaches for the calculation of the acoustic streaming around a spherical object were based on the assumption that the object is smaller than the wavelength, which makes it possible to obtain the analytical expression of the acoustic streaming. Doinikov proposed a method to solve the acoustic streaming around a particle of arbitrary sie. In this method, the spherical Bessel Functions were used to decompose the governing equation based on the Bessel Function orders and then each decomposed equation was solved independently. The accuracy of this solution approach depends on the total number of the orders involved. By using this method, the acoustic streaming around a gas bubble and an encapsulated particle were investigated. 2,3 Although this method is powerful, enabling the calculation of the acoustic streaming around a particle of arbitrary sie, it requires a long computational time and its results are not accurate in the farfield of liquid droplets as shown below. In this article, the Doinikov s approach has been improved by decomposing the acoustic streaming into compressional and shear components depending on the inside and outside of the viscous boundary layer around a microparticle of arbitrary sie to accurately and computationally efficiently calculate the acoustic streaming. Then, the governing equations of the acoustic streaming are decomposed into the homogenous and particular ones for the calculation of the compressional and shear components, respectively. The decomposed governing equations are then solved separately. The total acoustic streaming is then obtained by combining both the compressional and shear components. In this article, the acoustic streaming results obtained by using the proposed algorithm around a solid elastic particle and a liquid droplet are compared with those calculated by using the Doinikov s method. 2 CALCULATION OF ACOUSTIC STREAMING The proposed method for the calculation of the acoustic streaming generated around a microparticle/cell under an ultrasonic wave excitation is outlined in Fig.. Based on the linear oscillation solution of the fluid medium around a particle,4,5, the acoustic streaming is solved from the second-order time-independent governing equations when the ultrasonic excitation is at a frequency of ω. The interaction of the particle with the incident wave generates a thin viscous boundary layer δ v = (2η/ρ 0 /ω) /2, around the particle s surface, where η is the dynamic viscosity of the fluid and ρ 0 is the fluid density. Inside this layer, the spatial change rate of velocities is
3 significant and thus the fluid viscosity is dominant, while the fluid medium can be inviscid outside this layer. Based on this consideration, the acoustic streaming is decomposed based on the viscous boundary layer. Fig. - Framework for calculation of acoustic streaming generated around microparticle/cell under ultrasonic wave excitation. 2. Decomposition of Acoustic Streaming Governing Equations The first-order acoustic particle velocity vector v can be decomposed into the first-order scalar potential ϕ and vector potential Ψ as v v v, (a) v, (b) v Ψ, (c) where ϕ includes the incident and scattered scalar potential and Ψ is purely the scattered vector potential, and the corresponding first-order velocities are v ϕ and v ψ, respectively. The scalar potential ϕ is related to acoustic pressure (or compressional) waves, that are present inside and outside the viscous boundary layer. For a pressure wave, the fluid viscosity can be ignored in a finite domain (e.g., within a few wavelengths), and thus the corresponding velocity v ϕ is considered as an inviscid acoustic particle velocity in the nearfield of the spherical particle surface. On the contrary, the vector potential Ψ represents a shear wave and is only significant in the vicinity of the spherical particle surface where the spatial change rate of velocities is not negligible. The shear velocity vector v ψ is dominant in the range of a few δ v from the particle surface. The governing equation for the acoustic streaming are expressed as 6
4 v v v20 v v v v, (2a), (2b) where ρ is the first-order fluid density, v 20 is the acoustic streaming velocity vector, and represents the time-averaging of its inner term over one period from 0 to 2π/ω. By substituting the decomposed, first-order acoustic particle velocity in Eqn. (a) into Eqn. (2), the excitation sources of the acoustic streaming, i.e., the right-hand side (RHS) of Eqn. (2), can also be decomposed into two components: one source is purely related to v ϕ, and the other, to v ψ and the product of v ϕ and v ψ : i.e., v20h v, (3a) v20h v vv v, (3b) v20 p v, (4a) v20 p v v v vv v v v. (4b) In Eqns. (3) and (4), the two excitation sources decompose the acoustic streaming velocity into two components, v 20 = v 20h + v 20p where the subscript h represents the velocity component associated with v ϕ, and the subscript p represents the components related to v ψ and the product of v ϕ and v ψ : i.e., v20 v20h v 20 p, (5a) v20h 20h Ψ20h (5b) v Ψ. (5c) 20 p 20 p 20 p As shown in the following section, it is proved that the excitation source related to v ϕ becomes ero, resulting in that v 20h and v 20p are the homogeneous and particular solutions of Eqn. (2), respectively. Considering the effective domains of the individual excitation sources, it is concluded that v 20h exists inside and outside the viscous boundary layer while v 20p is only significant in the vicinity of the particle surface. 2.2 Homogeneous Acoustic Streaming Solution Vector: v 20h As discussed previously, the first-order acoustic particle velocity v ϕ associated with the scalar velocity potential is inviscid, leading to the fact that the pressure wave number k is a real number. Then, the inviscid Euler s Equation and the Helmholt Equation can be represented, respectively, as 2 ˆ ˆ i0k, (6) 2 2 ˆ ˆ k, (7) where the hat ^ represents a complex amplitude in frequency domain.
5 By plugging Eqns. (b), (6), and (7) into the left-hand side (LHS) of Eqn. (3a), the LHS is simplified as v v v ik ˆ ˆ * ik ˆ 2 ˆ * Re ( ) Re ( ) 2 2, (8) 2 4 ik ˆ ˆ * ik ˆ ˆ * Re ( ) ( ) Re ( )( ) where the asterisk * represents the complex conjugate. Therefore, Eqn. (3a) can be rewritten as v20h 0. (9) Then, the velocity vector v 20h can be expressed only by the vector potential: i.e, v Ψ, (0) 20h 20h which satisfies Eqn. (9). That is, the homogeneous scalar potential 20h can be a constant: i.e., 20h constant. () By plugging Eqns. (b), (6), and (7) into the RHS of Eqn. (3b), the first term of the RHS is simplified as v v vv v v * * Re ˆ ˆ v Re ˆ ˆ v 2 2 v v. (2) * 2 Re 0 k ˆ ˆ 2 0 For the second term of RHS in Eqn. (3b), the vector identity of (A)A = (AA)/2 - A ( A) can be applied as v v v v v v 0. (3) 2 By plugging Eqns. (2) and (3) into Eqn. (3b), Eqn. (3b) can be rewritten as 2 v 0. (4) Therefore, v 20h is the homogeneous solution of Eqn. (2). The governing equation for the homogeneous vector potential Ψ 20h is then derived from Eqns. (0) and (4) as 2 2 Ψ 0, (5) which is a homogeneous Biharmonic Equation. In the axisymmetric case, the solution to Eqn. (5) can be represented as 20h 20h
6 n n n n2 C C C C P cos Ψ, (6) 20h n 2n 3n 4n n n0 where = r/a, a is the radius of the particle, r and θ are the radial distance and polar angle in the spherical coordinate system, respectively, and (cosθ) is the associated Legendre Polynomials. By applying the boundary condition of Ψ 20h 0 as to Eqn. (6), the coefficients of C 3n and C 4n should be ero: i.e., C, C4 0. (7) 3n 0 The coefficients of C n and C 2n will be determined from the boundary conditions on the particle surface in the following section. Eqn. (6) describes the analytical acoustic streaming outside the viscous boundary layer. Since the coefficients C n and C 2n are constant, the outer acoustic streaming can be calculated in a relatively short computational time. n 2.3 Particular Acoustic Streaming Solution: v 20p The particular acoustic streaming solution vector v 20p consisting of the scalar potential ϕ 20p and vector potential Ψ 20p in Eqn. (5c) can be found by solving Eqns. (4a) and (4b), respectively, using a modified Doinikov s method, although the original Doinikov s method was used to solve the total solution. The basic idea of the Doinikov s method is to decompose the excitation source into a serial of Legendre polynomials and then solve the decomposed equations order by order using the orthogonality of these polynomials. Note that the velocity v 20p is only significant in the vicinity of the particle surface. For the implementation of this method, at r = a+0δ v, v 20p is significantly small and can be ignored, which provides a boundary condition for ϕ 20p and Ψ 20p. In addition, the product of v ψ and any other variable can be ignored at r > a+0δ v. Therefore, all the integrals in calculating the particular solution vector v 20p are accurate enough in the range of [a a+0δ v ], rather than the limit of [a, ] that was originally proposed by Doinikov. This short integral range reduces the computational time significantly. Here, is then defined as 0 v a. (8) Based on the procedure proposed by Doinikov, the solutions of the particular streaming potential ϕ 20 and Ψ 20p are expressed with the Legendre polynomials and the associated Legendre polynomials as Ψ 20 p 20 pn n n0 P cos, (9a) e P cos, (9b) 20 p 20 pn n n where n 20 pn y n ydyc5n y n ydyc6n n 2 n n, (9c)
7 n n3 20 pn y rn yn3 n ydyc9n 22n 3 n n C 7n y rn yn n ydy 22n n, (9d) n n2 y rn yn2 n ydyc0n 22n n2 n C 8n y rn yn n ydy 22n3 n μ n (), χ rn () and χ θn () are defined in Ref. By applying the boundary condition of ϕ 20p 0 and Ψ 20p 0 as, the coefficients of C 5n, C 6n, C 7n, C 8n, C 9n, and C 0n become eros: i.e., C, C6 0, C7 0, C8 0, C9 0, and C0 0. (20) 5n 0 n n By applying the second-order time-independent boundary conditions, the coefficients of C n and C 2n are determined as n n 2n a Cn n r20pn 20pn 2 nn nn ra ra, (2a) 2nn 20 pn ar20 pn 20 pn 2 ra ra ra n2 2n a C2n n r20pn 20pn 2 nn nn ra ra. (2b) 2nn 20 pn ar20 pn 20 pn 2 ra ra ra 3 SIMULATED ACOUSTIC STREAMING The proposed method can be used to calculate the acoustic streaming in the outer boundary layer analytically while the streaming in the inner boundary layer can be obtained by solving the integrals for the short integral range. For the simulated acoustic streaming results, two kinds of spherical particles are used in this paper. One is a polystyrene (PS) bead and the other is a liquid droplet, and the radii of both the particles are a = 0 μm. For the PS bead, its Young s modulus and Poisson s ratio are set to E = 3.4 GPa and = 0.35, respectively. 7 For the liquid droplet, its density and surface tension are ρ 0 = 00 kg/m 3 and T 0 = 0.2 N/m, respectively. 8 The acoustic streaming around the PS bead for an incident plane wave of ka = 0.0 is calculated and then the computational times of both the proposed method and the Doinikov s method are compared. Since the proposed method is advantageous to the calculation of the acoustic streaming around particles of non-compact sie (i.e., ka ), the acoustic streaming for the PS bead and the liquid droplet are then compared for their computational accuracies under the excitation condition of ka =. n n
8 3. Acoustic Streaming around Compact Polystyrene Bead of ka = 0.0 Figure 2 shows the acoustic streaming in the vicinity on the surface of the PS bead in water when a plane ultrasonic wave at the excitation frequency of ka = 0.0 is incident to the PS bead in the vertical direction (i.e., = 90). In Fig. 2, the four vortexes are observed and are symmetric with respect to the axes of x = 0 and y = 0. The maximum steaming velocity occurs at the surface of the PS bead, which is due to the large spatial gradient of the shear fluid medium velocity. The velocity nodal line is formed approximately at r = a + δ v and the velocity directions are opposite at both the sides of the nodal line. In the far field away from the surface, the velocity is decreased significantly. Velocity nodal line Fig. 2 - Acoustic streaming around PS bead in water under ultrasonic plane wave excitation at frequency of ka = 0.0 in incident angle of 90. The acoustic streaming is symmetric with respect to x = 0 and y = 0 and one vortex is shown in the right figure. Under the same condition of ka = 0.0, the computational times of both the Doinikov s method and the proposed method are compared in Table. These two methods were implemented in MathWork Matlab 204b for Mac. The computation was performed in an Apple imac Late 203 equipped with 32 GB RAM (600MH) and one Intel Core i7 processor with the clock frequency of 3.5 GH. The last column in Table shows the spatially-averaged percentage difference between the streaming velocity distributions calculated by using the two methods. Here, the percentage difference is defined as E r v20x v20 x_ ref v20 y v20 y_ ref Max 2 2 v20 x_ ref v20 x_ ref 2 2, (22) where the acoustic streaming velocity calculated using the proposed method is used as a reference. As shown in Table, the Doinikov s method results in smaller difference when the integral range is larger than 50a although its computational time is getting much longer. Obviously, the proposed method can produce the more accurate results in much shorter computational time than the Doinikov s method. Even for the integral range of [a 00a], the computational time of the
9 Doinikov s method is approximately 24.2 times longer than the proposed method. Therefore, the proposed method can be used to reduce the computational time significantly. Table - Comparison of computational time for PS bead for ka = 0.0. Doinikov s method Proposed method Spatially-Averaged Integral range Computational time [s] Integral range Computational time [s] Percentage Difference of Acoustic Streaming [a 00a] % [a 50a] % [a 200a] % [a a+0δ v ] 3.93 [a 250a] % [a 300a] % [a 350a] % 3.2 Acoustic Streaming around Polystyrene Bead of ka = For an incident wave of ka, the viscous boundary layer thickness δ v is relatively thin compared with microparticle radius a. In this simulation, the incident wavenumber is set to ka =. Then, the viscous boundary layer is δ v = 0.2 μm for a particle of 0 μm in radius. In this extremely thin layer, it is difficult to clearly observe the inner streaming either in a simulation or in an experiment. Thus, the outer streaming is only of interest. In this paper, the outer streaming for r > a + 4δ v is presented. Figure 3(a) shows the outer acoustic streaming generated around the spherical PS bead under an ultrasonic plane wave excitation of ka =. In Fig. 3(a), twelve vortexes are observed and these vortexes are symmetric with respect to the axes of x = 0 and y = 0. For the outer streaming, the maximum speed occurs in the vicinity of the top and bottom surface of the particle, different from the case of ka = 0.00 in Fig. 2. Figure 3(b) shows the percentage difference between the acoustic streaming results obtained by using the Doinikov s method and the proposed method (see Eqn. (39)). In Fig. 3(b), the maximum percentage difference is approximately 5.5%. In particular, the difference is increased along the red line shown in Fig. 3(b). Along this red line, the total streaming velocities (v 20 = v 20h + v 20p ) in the r- direction are compared in Fig. 3(c). Here, the streaming velocity in the viscous boundary layer is also included. For r <.5a, the total acoustic streaming velocity is much higher than that in the rest region. The velocities calculated by using both the methods are also close to each other in this region. As the r-direction distance increases, the difference between the two methods increases slightly. For the PS bead, these results agree well with each other, indicating that the proposed method is valid. 3.3 Acoustic Streaming around Liquid Droplet of ka = By using the same incident wave of ka =, the acoustic streaming around the liquid droplet is studied. In Fig. 4(a), it seems that there are four vortexes but they are not observed clearly, in contrary to the case of the PS bead in Fig. 3(a). It is interesting that the streaming velocity of the liquid droplet is much lower than that of the PS bead: i.e., in Figs. 3(a) and 4(a), the maximum acoustic streaming speeds of the PS bead and the liquid droplet are 0.22 μm/s and 0.09 μm/s, respectively. Figure 4(b) shows the percentage difference of the streaming velocity distributions
10 calculated by using both the two methods. In contrast to the results for the PS bead in Fig. 3(b), there is the large percentage difference in the calculated streaming velocities with the maximum difference of 20% at r = 4a, along the red line in Fig. 4(b), as shown in Fig. 4(c). In the vicinity of the droplet surface at r <.a, the two acoustic steaming results are in line with each other, providing the validation of the proposed method. For r >.a, the solution of the Doinikov s method starts to diverge, while the acoustic streaming velocity of the proposed method converges to ero. Fig. 3 - Outer acoustic streaming around PS bead in water under ultrasonic plane wave excitation at frequency of ka = in incident angle of 90: (a) Acoustic streaming calculated by using proposed method for range of [a+4δ v, 4a], (b) Percentage difference of streaming velocities (referenced with the streaming velocity distribution calculated by using proposed method), and (c) Streaming velocities in r-direction along red line in Fig. 3(b). 4 CONCLUSION In this paper, the novel algorithm is proposed to compute the acoustic streaming around a cell/microparticle. By decomposing the acoustic streaming into the compressional and shear streaming components, the governing equations for each streaming component are solved
11 independently. Then, the total acoustic streaming is obtained by combining the two resulting streaming components. From the comparison of the computational times of both the Doinikov s method and the proposed method, it is shown that the computational time of the proposed method is approximately /33.8 of that of the Doinikov s method for the same acoustic streaming results. In addition, under the incident wave excitation of ka =, the acoustic streaming results, around the PS beads, obtained by using the Doinikov s method and the proposed method are almost identical with the spatially-averaged maximum difference of 5.5%, indicating that the proposed method is valid. For the liquid droplet under the ultrasonic excitation of ka =, the acoustic streaming calculated by using the Doinikov s method is divergent at the farfield of the liquid droplet, while the proposed method results in the convergent streaming velocity. Therefore, it can be concluded that the proposed method can provide accurate acoustic streaming results in a relatively short computational time. Fig. 4 - Acoustic streaming around liquid droplet in water under under ultrasonic plane wave excitation at frequency of ka = in incident angle of 90: (a) Acoustic streaming calculated by using proposed method for range of [a+4δ v, 4a], (b) Percentage difference of acoustic streaming velocities (referenced with the streaming velocity distribution calculated by using proposed method), and (c) Streaming velocities in r-direction along red line in Fig. 4(b).
12 5 ACKNOWLEDGEMENT This work has been sponsored by research grants from the National Science Foundation of USA (Grant No.: ECCS-23225). 5 RERERENCES. W. P. Raney, J. C. Corelli, and P. J. Westervelt, "Acoustical Streaming in the Vicinity of a Cylinder," J. Acoust. Soc. Am., 26(6), , (954). 2. J. Holtsmark, I. Johnsen, T. Sikkeland, and S. Skavlem, "Boundary Layer Flow near a Cylindrical Obstacle in an Oscillating Incompressible Fluid, " J. Acoust. Soc. Am., 26(), 26-39, (954). 3. C. A. Lane, "Acoustical Streaming in the Vicinity of a Sphere," J. Acoust. Soc. Am, 27(6), , (955). 4. W. L. Nyborg, "Acoustic Streaming near a Boundary," J. Acoust. Soc. Am, 30(4), , (958). 5. C.-Y. Wang, "The flow field induced by an oscillating sphere," J. Sound Vib., 2(3), , (965). 6. N. Riley, "On a sphere oscillating in a viscous liquid," Q. J. Mech. Appl. Math., 9(4), , (966). 7. C. P. Lee and T. G. Wang, "Near-boundary streaming around a small sphere due to two orthogonal standing waves," J. Acoust. Soc. Am, 85(3), , (989). 8. C. P. Lee and T. G. Wang, "Outer acoustic streaming," J. Acoust. Soc. Am, 88(5), , (990). 9. H. Zhao, S. S. Sadhal, and E. H. Trinh, "Internal circulation in a drop in an acoustic field," J. Acoust. Soc. Am, 06(6), , (999). 0. A. Y. Rednikov, H. Zhao, and S. S. Sadhal, "Steady streaming around a spherical drop displaced from the velocity antinode in an acoustic levitation field," Q. J. Mech. Appl. Math, 59(3), , (2006).. A. A. Doinikov, "Acoustic radiation pressure on a compressible sphere in a viscous fluid," J. Fluid Mech., 267, -22, (994). 2. A. A. Doinikov and A. Bouaka, "Acoustic microstreaming around a gas bubble," J. Acoust. Soc. Am, 27(2), , (200). 3. A. A. Doinikov and A. Bouaka, "Acoustic microstreaming around an encapsulated particle," J. Acoust. Soc. Am, 27(3), , (200). 4. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, , (96). 5. Y.-H. Pao and C.-C. Mow, Diffraction of Elastic Waves and Dynamic Stress Concentrations, Crane, Russak & Company Inc., , (973). 6. M.F. Hamilton and D.T. Blackstock, Nonlinear Acoustics, Academic Press, 25-37, , (998) P. V. Zinin, J. S. A. III, and V. M. Levin, "Mechanical resonances of bacteria cells," Phys. Rev. E, 72(6), (2005).
Transient, planar, nonlinear acoustical holography for reconstructing acoustic pressure and particle velocity fields a
Denver, Colorado NOISE-CON 013 013 August 6-8 Transient, planar, nonlinear acoustical holography for reconstructing acoustic pressure and particle velocity fields a Yaying Niu * Yong-Joe Kim Noise and
More informationGraduate School of Engineering, Kyoto University, Kyoto daigaku-katsura, Nishikyo-ku, Kyoto, Japan.
On relationship between contact surface rigidity and harmonic generation behavior in composite materials with mechanical nonlinearity at fiber-matrix interface (Singapore November 2017) N. Matsuda, K.
More informationSound Pressure Generated by a Bubble
Sound Pressure Generated by a Bubble Adrian Secord Dept. of Computer Science University of British Columbia ajsecord@cs.ubc.ca October 22, 2001 This report summarises the analytical expression for the
More informationF11AE1 1. C = ρν r r. r u z r
F11AE1 1 Question 1 20 Marks) Consider an infinite horizontal pipe with circular cross-section of radius a, whose centre line is aligned along the z-axis; see Figure 1. Assume no-slip boundary conditions
More informationFlow Field and Oscillation Frequency of a Rotating Liquid Droplet
Flow Field and Oscillation Frequency of a Rotating Liquid Droplet TADASHI WATANABE Center for Computational Science and e-systems Japan Atomic Energy Agency (JAEA) Tokai-mura, Naka-gun, Ibaraki-ken, 319-1195
More informationarxiv:physics/ v1 [physics.bio-ph] 20 Mar 2003
An Estimate of the Vibrational Frequencies of Spherical Virus Particles arxiv:physics/0303089v1 [physics.bio-ph] 0 Mar 003 L.H. Ford Department of Physics and Astronomy Tufts University, Medford, MA 0155
More informationNonlinear dynamics of lipid-shelled ultrasound microbubble contrast agents
Computational Methods in Multiphase Flow IV 261 Nonlinear dynamics of lipid-shelled ultrasound microbubble contrast agents A. A. Doinikov 1 & P. A. Dayton 2 1 Belarus tate University, Belarus 2 University
More informationHigher Orders Instability of a Hollow Jet Endowed with Surface Tension
Mechanics and Mechanical Engineering Vol. 2, No. (2008) 69 78 c Technical University of Lodz Higher Orders Instability of a Hollow Jet Endowed with Surface Tension Ahmed E. Radwan Mathematics Department,
More informationParametric Investigation of the Common Geometry Shapes for Added Mass Calculation
Parametric Investigation of the Common Geometry Shapes for Added Mass Calculation Afsoun Koushesh* and Jin Lee Department of Mechanical Engineering, American University of Sharjah, Sharjah, UAE *Corresponding
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationEffect of Liquid Viscosity on Sloshing in A Rectangular Tank
International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 Volume 5 Issue 8 ǁ August. 2017 ǁ PP. 32-39 Effect of Liquid Viscosity on Sloshing
More information5.1 Fluid momentum equation Hydrostatics Archimedes theorem The vorticity equation... 42
Chapter 5 Euler s equation Contents 5.1 Fluid momentum equation........................ 39 5. Hydrostatics................................ 40 5.3 Archimedes theorem........................... 41 5.4 The
More informationFundamentals of Fluid Dynamics: Waves in Fluids
Fundamentals of Fluid Dynamics: Waves in Fluids Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/ tzielins/ Institute
More informationChapter 6: Incompressible Inviscid Flow
Chapter 6: Incompressible Inviscid Flow 6-1 Introduction 6-2 Nondimensionalization of the NSE 6-3 Creeping Flow 6-4 Inviscid Regions of Flow 6-5 Irrotational Flow Approximation 6-6 Elementary Planar Irrotational
More informationarxiv: v2 [math-ph] 14 Apr 2008
Exact Solution for the Stokes Problem of an Infinite Cylinder in a Fluid with Harmonic Boundary Conditions at Infinity Andreas N. Vollmayr, Jan-Moritz P. Franosch, and J. Leo van Hemmen arxiv:84.23v2 math-ph]
More informationAEROELASTIC ANALYSIS OF COMBINED CONICAL - CYLINDRICAL SHELLS
Proceedings of the 7th International Conference on Mechanics and Materials in Design Albufeira/Portugal 11-15 June 2017. Editors J.F. Silva Gomes and S.A. Meguid. Publ. INEGI/FEUP (2017) PAPER REF: 6642
More informationREE Internal Fluid Flow Sheet 2 - Solution Fundamentals of Fluid Mechanics
REE 307 - Internal Fluid Flow Sheet 2 - Solution Fundamentals of Fluid Mechanics 1. Is the following flows physically possible, that is, satisfy the continuity equation? Substitute the expressions for
More informationThree-dimensional Acoustic Radiation Force on an Arbitrary Located Elastic Sphere
Proceedings of the Acoustics Nantes Conference 3-7 April, Nantes, France Three-dimensional Acoustic Radiation Force on an Arbitrary Located Elastic Sphere D. Baresch a, J.-L. Thomas a and R. Marchiano
More informationarxiv: v1 [physics.flu-dyn] 21 Jan 2015
January 2015 arxiv:1501.05620v1 [physics.flu-dyn] 21 Jan 2015 Vortex solutions of the generalized Beltrami flows to the incompressible Euler equations Minoru Fujimoto 1, Kunihiko Uehara 2 and Shinichiro
More informationTransactions on Modelling and Simulation vol 10, 1995 WIT Press, ISSN X
The receptance method applied to the free vibration of a circular cylindrical shell filled with fluid and with attached masses M. Amabili Dipartimento di Meccanica, Universita di Ancona, 1-60131 Ancona,
More information1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem
1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem Objective of the Chapter: Formulation of the potential flow around the hull of a ship advancing and oscillationg in waves Results of the Chapter:
More informationRadial Growth of a Micro-Void in a Class of. Compressible Hyperelastic Cylinder. Under an Axial Pre-Strain *
dv. Theor. ppl. Mech., Vol. 5, 2012, no. 6, 257-262 Radial Growth of a Micro-Void in a Class of Compressible Hyperelastic Cylinder Under an xial Pre-Strain * Yuxia Song, Datian Niu and Xuegang Yuan College
More informationPhys101 Lectures 28, 29. Wave Motion
Phys101 Lectures 8, 9 Wave Motion Key points: Types of Waves: Transverse and Longitudinal Mathematical Representation of a Traveling Wave The Principle of Superposition Standing Waves; Resonance Ref: 11-7,8,9,10,11,16,1,13,16.
More informationFluid Dynamics Problems M.Sc Mathematics-Second Semester Dr. Dinesh Khattar-K.M.College
Fluid Dynamics Problems M.Sc Mathematics-Second Semester Dr. Dinesh Khattar-K.M.College 1. (Example, p.74, Chorlton) At the point in an incompressible fluid having spherical polar coordinates,,, the velocity
More informationFREE VIBRATION ANALYSIS OF THIN CYLINDRICAL SHELLS SUBJECTED TO INTERNAL PRESSURE AND FINITE ELEMENT ANALYSIS
FREE VIBRATION ANALYSIS OF THIN CYLINDRICAL SHELLS SUBJECTED TO INTERNAL PRESSURE AND FINITE ELEMENT ANALYSIS J. Kandasamy 1, M. Madhavi 2, N. Haritha 3 1 Corresponding author Department of Mechanical
More informationPhysics Dec Time Independent Solutions of the Diffusion Equation
Physics 301 10-Dec-2004 33-1 Time Independent Solutions of the Diffusion Equation In some cases we ll be interested in the time independent solution of the diffusion equation Why would be interested in
More informationAEROELASTIC ANALYSIS OF SPHERICAL SHELLS
11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V) 6th European Conference on Computational Fluid Dynamics (ECFD VI) E. Oñate, J. Oliver
More informationIntroduction to Marine Hydrodynamics
1896 1920 1987 2006 Introduction to Marine Hydrodynamics (NA235) Department of Naval Architecture and Ocean Engineering School of Naval Architecture, Ocean & Civil Engineering First Assignment The first
More informationDifferential criterion of a bubble collapse in viscous liquids
PHYSICAL REVIEW E VOLUME 60, NUMBER 1 JULY 1999 Differential criterion of a bubble collapse in viscous liquids Vladislav A. Bogoyavlenskiy* Low Temperature Physics Department, Moscow State University,
More informationSummary PHY101 ( 2 ) T / Hanadi Al Harbi
الكمية Physical Quantity القانون Low التعريف Definition الوحدة SI Unit Linear Momentum P = mθ be equal to the mass of an object times its velocity. Kg. m/s vector quantity Stress F \ A the external force
More informationAcoustic radiation by means of an acoustic dynamic stiffness matrix in spherical coordinates
Acoustic radiation by means of an acoustic dynamic stiffness matrix in spherical coordinates Kauê Werner and Júlio A. Cordioli. Department of Mechanical Engineering Federal University of Santa Catarina
More informationSurface Waves and Free Oscillations. Surface Waves and Free Oscillations
Surface waves in in an an elastic half spaces: Rayleigh waves -Potentials - Free surface boundary conditions - Solutions propagating along the surface, decaying with depth - Lamb s problem Surface waves
More informationMedical Imaging Injecting Mathematics into the Problem of Bubbly Blood. Sarah McBurnie Prof Jon Chapman OCIAM, University of Oxford
Medical Imaging Injecting Mathematics into the Problem of Bubbly Blood Sarah McBurnie Prof Jon Chapman OCIAM, University of Oxford Diagnostic Ultrasound http://www.polhemus.com/ 03/02/2009 2 Ultrasound
More informationContinuum Mechanics Lecture 5 Ideal fluids
Continuum Mechanics Lecture 5 Ideal fluids Prof. http://www.itp.uzh.ch/~teyssier Outline - Helmholtz decomposition - Divergence and curl theorem - Kelvin s circulation theorem - The vorticity equation
More informationFigure 3: Problem 7. (a) 0.9 m (b) 1.8 m (c) 2.7 m (d) 3.6 m
1. For the manometer shown in figure 1, if the absolute pressure at point A is 1.013 10 5 Pa, the absolute pressure at point B is (ρ water =10 3 kg/m 3, ρ Hg =13.56 10 3 kg/m 3, ρ oil = 800kg/m 3 ): (a)
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part IB Thursday 7 June 2007 9 to 12 PAPER 3 Before you begin read these instructions carefully. Each question in Section II carries twice the number of marks of each question in Section
More informationPredicting Breakup Characteristics of Liquid Jets Disturbed by Practical Piezoelectric Devices
ILASS Americas 2th Annual Conference on Liquid Atomization and Spray Systems, Chicago, IL, May 27 Predicting Breakup Characteristics of Liquid Jets Disturbed by Practical Piezoelectric Devices M. Rohani,
More informationThe effect of rigidity on torsional vibrations in a two layered poroelastic cylinder
Int. J. Adv. Appl. Math. and Mech. 3(1) (2015) 116 121 (ISSN: 2347-2529) Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics The effect of rigidity on
More informationDIFFRACTION OF PLANE SH WAVES BY A CIRCULAR CAVITY IN QUARTER-INFINITE MEDIUM
11 th International Conference on Vibration Problems Z. Dimitrovová et al. (eds.) Lisbon, Portugal, 9-12 September 2013 DIFFRACTION OF PLANE SH WAVES BY A CIRCULAR CAVITY IN QUARTER-INFINITE MEDIUM Hasan
More informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More informationAE301 Aerodynamics I UNIT B: Theory of Aerodynamics
AE301 Aerodynamics I UNIT B: Theory of Aerodynamics ROAD MAP... B-1: Mathematics for Aerodynamics B-: Flow Field Representations B-3: Potential Flow Analysis B-4: Applications of Potential Flow Analysis
More information1. Comparison of stability analysis to previous work
. Comparison of stability analysis to previous work The stability problem (6.4) can be understood in the context of previous work. Benjamin (957) and Yih (963) have studied the stability of fluid flowing
More informationGeneral introduction to Hydrodynamic Instabilities
KTH ROYAL INSTITUTE OF TECHNOLOGY General introduction to Hydrodynamic Instabilities L. Brandt & J.-Ch. Loiseau KTH Mechanics, November 2015 Luca Brandt Professor at KTH Mechanics Email: luca@mech.kth.se
More informationFluid Properties and Units
Fluid Properties and Units CVEN 311 Continuum Continuum All materials, solid or fluid, are composed of molecules discretely spread and in continuous motion. However, in dealing with fluid-flow flow relations
More informationOn spherical-wave scattering by a spherical scatterer and related near-field inverse problems
IMA Journal of Applied Mathematics (2001) 66, 539 549 On spherical-wave scattering by a spherical scatterer and related near-field inverse problems C. ATHANASIADIS Department of Mathematics, University
More informationMODELLING AND MEASUREMENT OF BACKSCATTERING FROM PARTIALLY WATER-FILLED CYLINDRICAL SHELLS
MODELLING AND MEASUREMENT OF BACKSCATTERING FROM PARTIALLY WATER-FILLED CYLINDRICAL SHELLS Victor Humphrey a, Lian Sheng Wang a and Nisabha Jayasundere b a Institute of Sound & Vibration Research, University
More informationNew Developments of Frequency Domain Acoustic Methods in LS-DYNA
11 th International LS-DYNA Users Conference Simulation (2) New Developments of Frequency Domain Acoustic Methods in LS-DYNA Yun Huang 1, Mhamed Souli 2, Rongfeng Liu 3 1 Livermore Software Technology
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationElectrodynamics Qualifier Examination
Electrodynamics Qualifier Examination August 15, 2007 General Instructions: In all cases, be sure to state your system of units. Show all your work, write only on one side of the designated paper, and
More informationLecture 9 Laminar Diffusion Flame Configurations
Lecture 9 Laminar Diffusion Flame Configurations 9.-1 Different Flame Geometries and Single Droplet Burning Solutions for the velocities and the mixture fraction fields for some typical laminar flame configurations.
More informationResearch Article A Study on the Scattering Energy Properties of an Elastic Spherical Shell in Sandy Sediment Using an Improved Energy Method
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 217, Article ID 9471581, 7 pages http://dx.doi.org/1.5/217/9471581 Publication Year 217 Research Article A Study on the Scattering
More informationOPAC102. The Acoustic Wave Equation
OPAC102 The Acoustic Wave Equation Acoustic waves in fluid Acoustic waves constitute one kind of pressure fluctuation that can exist in a compressible fluid. The restoring forces responsible for propagating
More informationStructural Acoustics Applications of the BEM and the FEM
Structural Acoustics Applications of the BEM and the FEM A. F. Seybert, T. W. Wu and W. L. Li Department of Mechanical Engineering, University of Kentucky Lexington, KY 40506-0046 U.S.A. SUMMARY In this
More informationLEAST-SQUARES FINITE ELEMENT MODELS
LEAST-SQUARES FINITE ELEMENT MODELS General idea of the least-squares formulation applied to an abstract boundary-value problem Works of our group Application to Poisson s equation Application to flows
More information2 D.D. Joseph To make things simple, consider flow in two dimensions over a body obeying the equations ρ ρ v = 0;
Accepted for publication in J. Fluid Mech. 1 Viscous Potential Flow By D.D. Joseph Department of Aerospace Engineering and Mechanics, University of Minnesota, MN 55455 USA Email: joseph@aem.umn.edu (Received
More informationTHEORY AND DESIGN OF HIGH ORDER SOUND FIELD MICROPHONES USING SPHERICAL MICROPHONE ARRAY
THEORY AND DESIGN OF HIGH ORDER SOUND FIELD MICROPHONES USING SPHERICAL MICROPHONE ARRAY Thushara D. Abhayapala, Department of Engineering, FEIT & Department of Telecom Eng, RSISE The Australian National
More informationGeneral Solution of the Incompressible, Potential Flow Equations
CHAPTER 3 General Solution of the Incompressible, Potential Flow Equations Developing the basic methodology for obtaining the elementary solutions to potential flow problem. Linear nature of the potential
More informationMETHODS OF THEORETICAL PHYSICS
METHODS OF THEORETICAL PHYSICS Philip M. Morse PROFESSOR OF PHYSICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY Herman Feshbach PROFESSOR OF PHYSICS MASSACHUSETTS INSTITUTE OF TECHNOLOGY PART II: CHAPTERS 9
More informationNatural frequency analysis of fluid-conveying pipes in the ADINA system
Journal of Physics: Conference Series OPEN ACCESS Natural frequency analysis of fluid-conveying pipes in the ADINA system To cite this article: L Wang et al 2013 J. Phys.: Conf. Ser. 448 012014 View the
More informationModeling of Ultrasonic Near-Filed Acoustic Levitation: Resolving Viscous and Acoustic Effects
Modeling of Ultrasonic Near-Filed Acoustic Levitation: Resolving Viscous and Acoustic Effects I. Melikhov *1, A. Amosov 1, and S. Chivilikhin 2 1 Corning Scientific Center, Russia, 2 ITMO University, Russia
More informationHydroelastic vibration of a rectangular perforated plate with a simply supported boundary condition
Fluid Structure Interaction and Moving Boundary Problems IV 63 Hydroelastic vibration of a rectangular perforated plate with a simply supported boundary condition K.-H. Jeong, G.-M. Lee, T.-W. Kim & J.-I.
More informationIMPROVED STRUCTURE-ACOUSTIC INTERACTION MODELS, PART II: MODEL EVALUATION Guseong-dong, Yuseong-gu, Daejeon Republic of Korea
ICSV14 Cairns Australia 9-12 July, 2007 IMPROVED STRUCTURE-ACOUSTIC INTERACTION MODELS, PART II: MODEL EVALUATION Abstract Moonseok Lee 1 *, Youn-Sik Park 1, Youngjin Park 1, K.C. Park 2 1 NOVIC, Department
More informationSEG/New Orleans 2006 Annual Meeting
Sergey Ziatdinov*, St. Petersburg State University, Andrey Bakulin, Shell International Exploration and Production Inc, Boris Kashtan, St. Petersburg State University Summary We investigate the influence
More informationLiquid Jet Breakup at Low Weber Number: A Survey
International Journal of Engineering Research and Technology. ISSN 0974-3154 Volume 6, Number 6 (2013), pp. 727-732 International Research Publication House http://www.irphouse.com Liquid Jet Breakup at
More informationKEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY OSCILLATIONS AND WAVES PRACTICE EXAM
KEELE UNIVERSITY PHYSICS/ASTROPHYSICS MODULE PHY-10012 OSCILLATIONS AND WAVES PRACTICE EXAM Candidates should attempt ALL of PARTS A and B, and TWO questions from PART C. PARTS A and B should be answered
More informationChapter 5. The Differential Forms of the Fundamental Laws
Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations
More informationMECHANICAL PROPERTIES OF FLUIDS:
Important Definitions: MECHANICAL PROPERTIES OF FLUIDS: Fluid: A substance that can flow is called Fluid Both liquids and gases are fluids Pressure: The normal force acting per unit area of a surface is
More informationSecond-order sound field during megasonic cleaning of patterned silicon wafers: Application to ridges and trenches
JOURNAL OF APPLIED PHYSICS VOLUME 90, NUMBER 8 15 OCTOBER 2001 Second-order sound field during megasonic cleaning of patterned silicon wafers: Application to ridges and trenches P. A. Deymier a) Department
More informationSound Waves. Sound waves are longitudinal waves traveling through a medium Sound waves are produced from vibrating objects.
Sound Waves Sound waves are longitudinal waves traveling through a medium Sound waves are produced from vibrating objects Introduction Sound Waves: Molecular View When sound travels through a medium, there
More informationVortex motion. Wasilij Barsukow, July 1, 2016
The concept of vorticity We call Vortex motion Wasilij Barsukow, mail@sturzhang.de July, 206 ω = v vorticity. It is a measure of the swirlyness of the flow, but is also present in shear flows where the
More informationWhy bouncing droplets are a pretty good model for quantum mechanics
Why bouncing droplets are a pretty good model for quantum mechanics Robert Brady and Ross Anderson University of Cambridge robert.brady@cl.cam.ac.uk ross.anderson@cl.cam.ac.uk Cambridge, June 2014 Robert
More information6.1 Momentum Equation for Frictionless Flow: Euler s Equation The equations of motion for frictionless flow, called Euler s
Chapter 6 INCOMPRESSIBLE INVISCID FLOW All real fluids possess viscosity. However in many flow cases it is reasonable to neglect the effects of viscosity. It is useful to investigate the dynamics of an
More information19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007
19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, -7 SEPTEMBER 007 ULTRASONIC PARTICLE MANIPULATION DEVICES FORMED BY RESONANTLY-EXCITED, CYLINDRICAL STRUCTURES PACS: 43.5.Uv Kaduchak, Gregory 1 ; Ward,
More informationUltrasonic particle and cell separation and size sorting
SMR.1670-25 INTRODUCTION TO MICROFLUIDICS 8-26 August 2005 Ultrasonic Particle and Cell Separation and Size Sorting in Micro-channels V. Steinberg Weizmann Institute of Science, Israel Ultrasonic particle
More informationCHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION
CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,
More informationPressure corrected SPH for fluid animation
Pressure corrected SPH for fluid animation Kai Bao, Hui Zhang, Lili Zheng and Enhua Wu Analyzed by Po-Ram Kim 2 March 2010 Abstract We present pressure scheme for the SPH for fluid animation In conventional
More informationChapter 9: Differential Analysis
9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control
More informationProducing a Sound Wave. Chapter 14. Using a Tuning Fork to Produce a Sound Wave. Using a Tuning Fork, cont.
Producing a Sound Wave Chapter 14 Sound Sound waves are longitudinal waves traveling through a medium A tuning fork can be used as an example of producing a sound wave Using a Tuning Fork to Produce a
More information. (70.1) r r. / r. Substituting, we have the following equation for f:
7 Spherical waves Let us consider a sound wave in which the distribution of densit velocit etc, depends only on the distance from some point, ie, is spherically symmetrical Such a wave is called a spherical
More informationUNIVERSITY of LIMERICK
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 2012-13 MODULE TITLE: Introduction to Fluids DURATION OF
More information2.5 Stokes flow past a sphere
Lecture Notes on Fluid Dynamics.63J/.J) by Chiang C. Mei, MIT 007 Spring -5Stokes.tex.5 Stokes flow past a sphere Refs] Lamb: Hydrodynamics Acheson : Elementary Fluid Dynamics, p. 3 ff One of the fundamental
More informationSound Transmission in an Extended Tube Resonator
2016 Published in 4th International Symposium on Innovative Technologies in Engineering and Science 3-5 November 2016 (ISITES2016 Alanya/Antalya - Turkey) Sound Transmission in an Extended Tube Resonator
More informationProceedings of Meetings on Acoustics
Proceedings of Meetings on Acoustics Volume 19, 13 http://acousticalsociety.org/ ICA 13 Montreal Montreal, Canada - 7 June 13 Structural Acoustics and Vibration Session 4aSA: Applications in Structural
More informationFormation and Long Term Evolution of an Externally Driven Magnetic Island in Rotating Plasmas )
Formation and Long Term Evolution of an Externally Driven Magnetic Island in Rotating Plasmas ) Yasutomo ISHII and Andrei SMOLYAKOV 1) Japan Atomic Energy Agency, Ibaraki 311-0102, Japan 1) University
More informationTwo-dimensional ternary locally resonant phononic crystals with a comblike coating
Two-dimensional ternary locally resonant phononic crystals with a comblike coating Yan-Feng Wang, Yue-Sheng Wang,*, and Litian Wang Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing,
More information12.1 Viscous potential flow (VPF)
1 Energy equation for irrotational theories of gas-liquid flow:: viscous potential flow (VPF), viscous potential flow with pressure correction (VCVPF), dissipation method (DM) 1.1 Viscous potential flow
More informationKelvin Helmholtz Instability
Kelvin Helmholtz Instability A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram November 00 One of the most well known instabilities in fluid
More informationPhysics in Faculty of
Why we study Physics in Faculty of Engineering? Dimensional analysis Scalars and vector analysis Rotational of a rigid body about a fixed axis Rotational kinematics 1. Dimensional analysis The ward dimension
More informationNumerical Studies of Droplet Deformation and Break-up
ILASS Americas 14th Annual Conference on Liquid Atomization and Spray Systems, Dearborn, MI, May 2001 Numerical Studies of Droplet Deformation and Break-up B. T. Helenbrook Department of Mechanical and
More informationNumerical study on scanning radiation acoustic field in formations generated from a borehole
Science in China Ser. G Physics, Mechanics & Astronomy 5 Vol.48 No. 47 56 47 Numerical study on scanning radiation acoustic field in formations generated from a borehole CHE Xiaohua 1, ZHANG Hailan 1,
More informationDetailed Outline, M E 521: Foundations of Fluid Mechanics I
Detailed Outline, M E 521: Foundations of Fluid Mechanics I I. Introduction and Review A. Notation 1. Vectors 2. Second-order tensors 3. Volume vs. velocity 4. Del operator B. Chapter 1: Review of Basic
More informationStatistical Analysis of the Effect of Small Fluctuations on Final Modes Found in Flows between Rotating Cylinders
Statistical Analysis of the Effect of Small Fluctuations on Final Modes Found in Flows between Rotating Cylinders Toshiki Morita 1, Takashi Watanabe 2 and Yorinobu Toya 3 1. Graduate School of Information
More informationINDEX 363. Cartesian coordinates 19,20,42, 67, 83 Cartesian tensors 84, 87, 226
INDEX 363 A Absolute differentiation 120 Absolute scalar field 43 Absolute tensor 45,46,47,48 Acceleration 121, 190, 192 Action integral 198 Addition of systems 6, 51 Addition of tensors 6, 51 Adherence
More informationChapter 9: Differential Analysis of Fluid Flow
of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known
More informationPressure corrections for viscoelastic potential flow analysis of capillary instability
ve-july29-4.tex 1 Pressure corrections for viscoelastic potential flow analysis of capillary instability J. Wang, D. D. Joseph and T. Funada Department of Aerospace Engineering and Mechanics, University
More informationModule 7: Micromechanics Lecture 29: Background of Concentric Cylinder Assemblage Model. Introduction. The Lecture Contains
Introduction In this lecture we are going to introduce a new micromechanics model to determine the fibrous composite effective properties in terms of properties of its individual phases. In this model
More informationDifferential Acoustic Resonance Spectroscopy Analysis of Fluids in Porous Media
http://ijopaar.com; 2016 Vol. 2(1); pp. 22-30 Differential Acoustic Resonance Spectroscopy Analysis of Fluids in Porous Media Dr.Mohammad Miyan Associate Professor, Department of Mathematics, Shia P.G.College,
More informationSimulation of SAW-Driven Microparticle Acoustophoresis Using COMSOL Multiphysics
Simulation of SAW-Driven Microparticle Acoustophoresis Using COMSOL Multiphysics Nitesh Nama 1, R. Barnkob 2, C. J. Kähler 2, F. Costanzo 1, and T. J. Huang 1 1 Department of Engineering Science and Mechanics
More informationIn this section, mathematical description of the motion of fluid elements moving in a flow field is
Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small
More informationGENERAL PHYSICS (3) LABORATORY PHYS 203 LAB STUDENT MANUAL
Haifaa altoumah& Rabab Alfaraj By Haifaa altoumah& Rabab Alfaraj GENERAL PHYSICS (3) LABORATORY PHYS 203 LAB STUDENT MANUAL Name:-. ID# KING ABDULAZIZ UNIVERSITY PHYSICS DEPARMENT 1st semester 1430H Contents
More information